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I first posed this question when I was a first year student. I came up with some ad hoc arguments as to why the result is true (a bit of numerical experimentation), but never had a proof. I forgot about it until the other day, and thought it was still an interesting question. First, some motivation.

Take the expression

$$\sum_{n=0}^\infty \frac{x^n}{n!}$$

which equals its own derivative, that much is straight forward. The $x^n$ gets sent to $n x^{n-1}$ and the factorial kills the $n$, and the same expression emerges.

We can do the same thing with the integral.

$$\int_{-\infty}^\infty \frac{x^y}{\Gamma(y+1)}\,dy$$

formally satisfies being its own derivative--albeit the expression does not converge (which is a shame). But if you act like it converges, it should converge to $e^x$ because it's its own derivative.

This led me to considering a more suitable case, where the integral actually converges, and considering if the two functions agree. For example,

$$F(x) = \int_{-\infty}^{\infty}e^{-y^2} \frac{x^y}{\Gamma(y+1)}\,dy$$

defines a $C^{\infty}$ function in $x$ for $x \in \mathbb{R}^+$. This function satisfies $F'(x) = e^{-1}F(\frac{x}{e^2})$.

If we take

$$G(x) = \sum_{n=0}^\infty e^{-n^2}\frac{x^n}{n!}$$

then equally so, $G'(x) = e^{-1}G(\frac{x}{e^2})$.

Does $G = F$?

I thought of this in a more general setting though. I wondered if it happened for $e^{-y^2}$, does it work for other functions?

It may be useful to explain the ad hoc arguments I had when I was younger. These arguments were a summation of identities which are satisfied by both expressions, which seems implausible if they weren't equal in some sense. I'll explain them in the obtuse language I used when I was a first year, as it is the best way I can think of framing it. Let $\mathcal{I}$ and $\mathcal{S}$ denote the following operators,

$$\mathcal{I} \{f\} (x) = \int_{-\infty}^\infty f(y)\frac{x^y}{\Gamma(y+1)}\,dy$$

$$\mathcal{S} \{f\} (x) = \sum_{n=0}^\infty f(n) \frac{x^n}{n!}$$

Also, let $Qf(x) = xf(x)$ and $Tf(x) = f(x+1)$. Then $\mathcal{S}$ and $\mathcal{I}$ behave identically under actions of the two operators. The following similar identities present evidence as to why these two operators may be equivalent in a good enough scenario.

$$\mathcal{I}\{Tf\}(x) = \frac{d}{dx}\mathcal{I} \{f\} (x)$$

$$\mathcal{S} \{Tf\} (x) = \frac{d}{dx}\mathcal{S} \{f\} (x)$$

$$\mathcal{I}\{Qf\}(x) = Q\mathcal{I}\{Tf\}(x) = Q\frac{d}{dx}\mathcal{I} \{f\} (x)$$ $$\mathcal{S} \{Qf\} (x) = Q\mathcal{S} \{Tf\} (x) = Q\frac{d}{dx}\mathcal{S} \{f\} (x)$$

From these we also get

$$Q \mathcal{I}\{f\}(x) = \mathcal{I}\{QT^{-1}f\}(x)$$ $$Q \mathcal{S}\{f\}(x) = \mathcal{S}\{QT^{-1}f\}(x)$$

From all this the question is simple,

Are these two operators ever equivalent, if so when?

Or phrased more poetically

Is the continuous Taylor series no different than the discrete Taylor series?

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  • $\begingroup$ You could consider the same integral with any $1$-periodic function of $y$ inserted. They all satisfy the same identity. Are they all equal, up to a constant multiple? I suspect not. $\endgroup$ – Will Sawin May 31 '17 at 2:30
  • $\begingroup$ It's natural to (sort of) rephrase this as: do we have uniqueness for the delay differential equation $y'(x)=cy(ax)$, and (I know nothing about this but) I think not in the sense that $y(1)$ determines $y$; rather you need $y$ on an interval. $\endgroup$ – Christian Remling May 31 '17 at 2:32
  • $\begingroup$ Also, $G$ is entire while $F$ looks like we have a real problem at $x=0$. $\endgroup$ – Christian Remling May 31 '17 at 2:33
  • $\begingroup$ More specifically, it's easy to prove that if $y,w$ are two solutions of $y'(x)=cy(ax)$ and they agree on $[a,1]$, then they are identical. I think that's all that is true here; in other words, the DDE determines the function not up to a multiplicative constant, but up to a function on an interval. $\endgroup$ – Christian Remling May 31 '17 at 2:37
  • $\begingroup$ @ChristianRemling I considered Delay differential equations but never got far (I was a first year undergrad though, maybe relooking into it would be a good idea). Also, $F$ is technically holomorphic on $\mathbb{C}/(-\infty,0)$ but $F$ could be holomorphic on $\mathbb{C}$ if the branch cut is 'removable.' If it isn't this would be a great way of showing the functions aren't equal! $\endgroup$ – user78249 May 31 '17 at 2:40
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No. In fact I will show the map from $1$-periodic signed measures $\mu$ to $$F_\mu(x) = \int_{-\infty}^\infty \mu(y) e^{-y^2}\frac{x^y}{\Gamma(y+1)}dy$$ is injective for "reasonable" signed measures $\mu$.

Indeed, consider what happens to the following expression as $y_0$ goes to $\infty$ while the residue of $y_0$ modulo $1$ remains fixed:

$$ \Gamma(y_0+1) e^{-y_0^2 - y_0 \operatorname{dlog} \Gamma(y_0+1) } F_{\mu} ( e^{ 2 y_0 + \operatorname{dlog} \Gamma(y_0+1)} ) $$

$$ = \int_{-\infty}^{\infty} \mu(y) e^{-y^2 + 2 y y_0 - y_0^2} \frac{ \Gamma(y_0+1) e^{ y \operatorname{dlog} \Gamma(y_0+1)- y_0 \operatorname{dlog} \Gamma(y_0+1)}}{\Gamma(y+1)} dy$$

$$ = \int_{\infty}^{\infty} \mu(y+y_0) e^{-y^2} \frac{ \Gamma(y_0+1) e^{ y \operatorname{dlog} \Gamma(y_0+1)}}{\Gamma(y+y_0+1)} dy$$

Because $\Gamma(y+1)$ is log-convex and its slope is roughly $\log y$, the exponential-linear approximation to the slope gets better as $y_0$ goes to $\infty$, and so this

$$ \approx \int_{-\infty}^{\infty} \mu(y+y_0) e^{-y^2} dy $$

uniquely determines the convolution of $\mu$ with the Gaussian. Now because none of the Fourier transform coefficients of the Gaussian vanish, we can extract all the Fourier series coefficients of $\mu$ from this, which under mild analytic conditions extracts $\mu$.

In particular, for your example $F(x)$ we see that $F(e^{2 y_0 + \operatorname{dlog} \Gamma(y_0+1)})$ is asymptotic to a constant times $\frac{ e^{y_0^2 + y_0 \operatorname{dlog} \Gamma(y_0+1) }}{ \Gamma(y_0+1)}$ while this is not true for your Taylor series $G(x)$.

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    $\begingroup$ Ha! That is great! This is actually more interesting! It's always cool when a false result makes the concept better and stronger. Personally, I interpret this to mean we can construct arbitrary "Taylor series" with any 1-periodic measure ($F_\mu(x) = \int...\,d\mu$) and they never equal, but they satisfy the same operator equations I wrote in the question. That is really cool, thanks a ton! Back to sifting through my first year notes... $\endgroup$ – user78249 May 31 '17 at 3:16
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    $\begingroup$ One small point: you say and probably mean "measures" (that is, not necessarily absolutely continuous), but you write $\mu(y)\, dy$ instead of $d\mu(y)$, which I find confusing. (The added generality of measures with a possible singular part is useful of course; for example, the power series corresponds to $\mu=\sum\delta_n$.) $\endgroup$ – Christian Remling Jun 1 '17 at 0:06
  • $\begingroup$ @ChristianRemling Eventually I just assumed he meant $d\mu$ but that also confused me at first. As a side note, I'm also thinking we may be able to prove this more rigorously by using the Fourier Transform. (Adding a variable $a$ and using a coefficient $e^{2\pi a yi}$). At least, I'm close to doing it not for measures, but with holomorphic 1-periodic functions. Not the best at measure theory, more at home with holomorphic functions. I'll answer my own question if it works out. $\endgroup$ – user78249 Jun 1 '17 at 7:14
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So I think I found a way of proving $F \neq G$ using distribution theory. I'm not the best at distribution theory, it's definitely not my field, but I think I'm doing something right. This is as much rigor as I can produce in distribution theory, but I think I'm on the right track. I don't have a referential brain for distribution theory, I'm going off what I can remember, I apologize if I'm not explicitly stating the theorems I'm using. This idea works generally as to why the continuous Taylor series can never equal the discrete Taylor series, but I'll be a bit cavalier. Maybe this argument works in more general cases, but one would need to know more about distribution theory to produce such a result, sadly I don't.

Let's just use $e^{-y^2}$ to be explanatory. If we take

$$F(a,x) = \int_{-\infty}^\infty e^{-y^2} e^{-2\pi a y i} \frac{x^y}{\Gamma(y+1)}\,dy$$

Then we can look at the equivalency in a Fourier sense.

Take the Poisson summation

$$\sum_{n=-\infty}^\infty F(n,x) = \sum _{n=-\infty}^\infty e^{-n^2} \frac{x^n}{n!} = \sum _{n=0}^\infty e^{-n^2} \frac{x^n}{n!}$$

which follows because $\frac{1}{n!} = 0$ for $n<0$. Now I'm not certain about the following interchange, but I'm fairly certain it should be allowed in good enough scenarios (especially with $e^{-y^2}$).

$$\sum_{n=-\infty}^\infty F(n,x) = \sum_{n=-\infty}^\infty \int_{-\infty}^\infty e^{-y^2} e^{-2\pi i n y} \frac{x^y}{\Gamma(y+1)}\,dy = \int_{-\infty}^\infty \chi(y)e^{-y^2} \frac{x^y}{\Gamma(y+1)}\,dy$$

where $\chi$ is the dirac comb. This is kind of nothing new to what Christian Remling said, except I'm phrasing it distributionally, which is to our advantage as we add the Fourier transform. As least, if I've not mistaken myself. The above expression can be thought of as an inner product. If the discrete and continuous Taylor series equal each other, then $(\chi, h) = (1,h)$ or $(\chi -1, h) =0$, where

$$h(y) = e^{-y^2}\frac{x^y}{\Gamma(y+1)}$$

I'm kind of out of my element right here, if I'm wrong, definitely correct me. Let $\mathcal{F}$ denote the Fourier Transform. Let $\delta$ denote the dirac $\delta$ function

$$\mathcal{F}\{\chi -1\} = \sum_{n=-\infty}^\infty e^{2\pi i n y} - \delta = \chi - \delta$$

satisfies

$$(\mathcal{F}\{\chi-1\},h) = \sum_{n=1}^\infty e^{-n^2}\frac{x^n}{n!} \neq 0$$

Notice we remove the $n=0$ term. This produces the final result. It is straight forward from here! We just use the unitary properties of $\mathcal{F}$ under this inner product.

$$(\chi-1,\mathcal{F}^{-1}h) \neq 0$$

Apply $\mathcal{F}$ again, using the unitary property again, and through the exact same argument

$$(\chi-1,h) = (\mathcal{F}\{\chi -1\}, \mathcal{F}^{-1}h) = (\chi - \delta, \mathcal{F}^{-1} h)\neq 0 $$

therefore

$$(\chi-1,h) \neq 0$$

The continuous Taylor series cannot equal the discrete Taylor series. Personally, I'm ashamed to ask, but for what functions does this actually work. I know this argument must work in a distributional sense, but for what functions does this actually work when written in a more classical sense?

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